1. Field of the Invention
The invention relates to the field of reflective dielectric structures, and more particularly to broadband reflective dielectric structures used as mirrors in laser systems.
2. Description of Related Art
Ultra short-pulse generation has advanced to a level where the bandwidth of standard Bragg mirrors, e.g. composed of TiO2 and SiO2 quarter-wave layers, limits the pulse width or tunability of the generated laser pulses. The limitation is two fold. First, due to the limited difference in refractive index of both materials, e.g., nTiO2≈2.4 and nSiO2≈1.45 the high reflectivity bandwidth of a standard quarter-wave Bragg mirror centered at 800 nm is only about 200 nm. Second, the higher order group delay dispersion (GDD) produced by quarter-wave Bragg mirrors further limits the useful bandwidth to about 100 nm which is just enough bandwidth for 10 fs pulses.
In a chirped mirror, the Bragg wavelength, xcexB, of the individual layer pairs is varied from layer pair to layer pair (e.g. linearly), so that longer wavelengths penetrate deeper into the mirror structure than shorter wavelengths before being reflected. Such mirrors show an enlarged high reflectivity range and show a negative dispersion. However, the dispersion properties of these mirrors may be inadequate for ultra short pulse generation.
Chirped mirrors are also beneficial for the compression of high energy pulses, because they produce high dispersion with little material in the beam path, thereby avoiding nonlinear effects in the compressor. Thus, the design of these mirrors is extremely important for the further development of ultra fast laser sources.
It turns out that the design of a chirped mirror does not necessarily lead to a smooth and controlled GDD of the mirror. Using standard transfer matrix analysis of the multilayer structure as discussed in xe2x80x9cExact coupled mode theories for multilayer interference coating with arbitrary strong index modulations,xe2x80x9d IEEE J. Quant. Elec., vol. 33, March 1997, which is hereby incorporated by reference, one observes that the group delay produced by such a chirped mirror does not vary linearly with wavelength, as one would expect for a mirror with linearly chirped Bragg wavelength. The local average of the group delay shows the expected tendency to increase linearly with increasing wavelength. However, it also exhibits strong oscillations. The cause of these oscillations is the following. Longer wavelengths have to pass the first section of the Bragg mirror, which acts as a transmission grating for these wavelengths. The slight reflection in the front section interferes with the strong reflections from the deeper layers, as in a Gires-Tournouis Interferometer (GTI). The oscillations in the group delay have an amplitude of several tens of femtoseconds, which make these simple-chirped mirrors less useful for ultra short pulse generation.
What is needed is a mirror design which reduces the oscillations in the group delay, allowing control of the group delay dispersion while maintaining broad band reflectivity and low group delay dispersion.
One embodiment of the present invention is a double-chirped mirror in which an initial design of a double chirped mirror is generated and then perturbed to improve the group delay dispersion and reflectance characteristics by accounting for the frequency dependence of design parameters such as indices of refraction. In the initial design, the oscillations in the group delay are avoided by varying not only the Bragg period or Bragg wavelength in the mirror, but also by tailoring the coupling between the forward and the backward propagating waves inside the mirror such that spurious reflections leading to GTI-effects are consistently avoided. In this embodiment of the invention both quantities, the Bragg wavelength and the coupling coefficient, are chirped. The oscillations in the group delay are reduced or eliminated by a sufficiently slow increase in the coupling of the wave incident onto the mirror and the wave reflected from the mirror as the incident wave moves through the first sections of the mirror. This slow increase in the coupling of the waves, to avoid spurious reflection in the front section of the mirror, is provided by the matching sections.
In this embodiment of the invention, the mismatch and therefore the oscillations in the group delay are reduced by addressing two of the matching problems encountered in a standard chirped mirror. First, the medium from which the radiation is incident on the mirror is matched to the first layer of a second matching section mirror by a first matching section, which is, for example, provided by a high quality broadband antireflection coating. The antireflection coating can be designed with commercially available dielectric coating design programs. The second matching section has to match from the antireflection coating section to a simple chirped section. In this way, the double-chirped mirror is generated with a controlled group delay and an extended high reflectivity range when compared to standard dielectric Bragg mirrors. This analytic starting structure helps to avoid internal resonances in the multilayer structure.
In this embodiment, the material parameters used to construct the initial design are assumed to be constant in frequency over the frequency range of interest. Once the initial design has been constructed, a subsequent computer based perturbation of the initial design can take into account the wavelength dependence of the refractive indices of the materials used to construct the mirror, and other design goals, such as a highly transmitting wavelength range close to the high reflection wavelength range of the mirror which may be useful for coupling a pump laser beam into a laser cavity.
In one embodiment of the invention, each of the sections of the mirror comprises sets of layers in a plurality of layers. Layers in the plurality of layers are composed of materials with a high or low index of refraction for a frequency range of electromagnetic radiation.
In one embodiment of the invention, the perturbation of the initial design is achieved by constructing a merit function including the reflectance, group delay, and/or group delay dispersion of the mirror, or any combination of these parameters. The merit function is then optimized by adjusting the thicknesses of layers in sections of the mirror until acceptable reflectance, group delay, and/or group delay dispersion are achieved. In other embodiments, the reflectance portion of the merit function includes reducing the reflectance of the mirror in a wavelength range which may be used to transmit a pump laser beam through the mirror. In another embodiment of the invention, the merit function is alternately optimized for reflectivity, group delay and/or group delay dispersion by varying weighting functions.
In still another embodiment, the initial design is perturbed to achieve a final design such that the GDD varies by less than 100% from its average value, but by more than 20% from its average value over more than a continuous half part of the high reflectivity band of the mirror. In yet another embodiment of the invention, the initial design is perturbed to achieve a final design such that the GDD varies by less than 200% from its average value, but by more than 20% from its average value over more than a continuous half part of the high reflectivity band of the mirror. The high reflectivity band of the mirror is defined to mean a continuous frequency range over which the reflectivity of the mirror is higher than 99%.
In still other embodiments the initial design is perturbed to achieve a final design such that the GDD varies by less than 200% from its average value over the high reflectivity band of the mirror, less than 100% from its average value over the high reflectivity band of the mirror, less than 50% from its average value over the high reflectivity band of the mirror, or less than 20% from its average value over the high reflectivity band of the mirror.
Yet another embodiment of the invention is a method of constructing a double chirped mirror for a frequency range of electromagnetic radiation, comprising specifying a design including a plurality of layers, the plurality of layers being transparent to the electromagnetic radiation and having refractive indices which vary between layers in the plurality of layers, and wherein for a first set of layers the optical thickness of alternate layers in the set of layers varies monotonically and the total optical thickness of a layer and the two adjacent half layers in the set of layers varies monotonically. The design is perturbed by adjusting the optical thickness of layers in the plurality of layers to account for the frequency dependence of the refractive indices.
In another embodiment, perturbing the design includes reducing the difference between the group delay dispersion for a plurality of frequencies in the frequency range of interest and a design goal group delay dispersion for the plurality of frequencies. Perturbing the design may also include reducing the difference between the reflectance for a plurality of frequencies in the frequency range of interest and a design goal reflectance for the plurality of frequencies. Furthermore, perturbing the design may also include constructing a function having as its input a design goal and a current design parameter, wherein the output of the function is related to a difference between the design goal and the current design parameter.
In yet another embodiment, specifying the design includes specifying a second set of layers in the plurality of layers, wherein the optical thickness of a layer is the same as the optical thickness of two adjacent half layers in the set of layers, and the total optical thickness of the layer and the two adjacent half layers in the second set of layers varies monotonically. Specifying the design may also include specifying a third set of layers in the plurality of layers, wherein the optical thickness of each layer in the third set of layers is substantially a quarter-wavelength thick for a frequency in the frequency range of electromagnetic radiation. Additionally, specifying the design includes specifying a fourth set of layers in the plurality of layers, wherein the fourth set of layers comprises an antireflection coating.
Still another embodiment of the invention includes a double chirped mirror as part of a laser cavity. In this embodiment the laser may be a short pulse laser, or an ultrashort pulse laser, including a femtosecond laser. Yet another embodiment of the invention includes a double chirped mirror as part of an amplifier. In this embodiment the amplifier may be a short pulse amplifier, or an ultrashort pulse amplifier, including a femtosecond pulse amplifier. In another aspect of this embodiment, the amplifier is a high power amplifier.